Welcome back to "Continuous Hit Music" as we march on in search of the inner workings of music. In the last article, we discovered what made two pitches the same note. What we haven't touched on yet is all of the notes in between. Fortunately for us, we have Julie Andrews to show us the way.
The Sound of Music is a Broadway musical-turned-movie that revolves around a free-spirited nun-in-training named Maria who ends up as the nanny to the children of an Austrian navy captain. During the film, at some point between melting Captain von Trapp's heart and hiding from Nazis, Maria (played by Dame Julie Andrews) sings the catchiest intro lesson to music theory I've ever heard, "Do-Re-Mi". Each of the syllables that she mentions in the song -- do, re, mi, fa, so, la, and ti -- is one of the seven notes in a major scale.
Something about this always threw me, though. For years I wondered, "why are the notes of a major scale those notes? Why are there seven notes instead of, say, ten? How did we find those exact notes?" Fortunately, the answers can all be found in physics.
In the first CHM article, we discovered that musical sounds tend to be made up of a fundamental frequency, along with additional "harmonic" frequencies, which are integer multiples of the fundamental. (For instance, a sound with a fundamental frequency of 200 Hz would also have harmonics of 400 Hz, 600 Hz, 800 Hz, 1000 Hz, and beyond, or 2*200, 3*200, 4*200, 5*200, and so on.) All harmonics are not created equal, however. Each harmonic doesn't sound at the same volume. Generally, the fundamental frequency is easily the loudest frequency in the sound, with the harmonics getting quieter and quieter as they get higher and higher. As a result, there is a much greater emphasis on lower harmonics, since you hear them far more loudly than the higher ones.
With that emphasis in mind, let's look at the lower harmonics of an A note. Using 220 Hz as our fundamental frequency, here is the fundamental and the first five harmonics: 220 Hz, 440 Hz, 660 Hz, 880 Hz, 1100 Hz, 1320 Hz. As we covered in the last CHM article, each time a frequency is doubled, it remains the same note. Since 220 Hz is an A, so are 440 Hz and 880 Hz, helping to reinforce the identity of the note. Similarly, 660 Hz and 1320 Hz must be the same note, since 660 multiplied by two is 1320.
But would this new note at 660 Hz be? If we were to plunk around on a keyboard to find it, it would be an E. In fact, if we were to find all of the frequencies listed on a piano, we would find them as the notes A, A, E, A, C# (said as "C-sharp"), and E. And if you started singing "Do-Re-Mi", starting with the note A as "do," you'd find that C# would be the note you'd sing at "mi," and E would be the note you'd sing at "so." That leaves us with "re," "fa," "la," and "ti" to find.
We can start by looking at the note "E." Since it's the first harmonic of A that isn't an A itself, it stands to reason that it's the note most closely related to A. So what does its harmonics look like? Let's start by dividing 660 Hz by two to keep the numbers manageable. (Remember, multiplying or dividing a fundamental frequency by two gives you the next higher or lower note of the same name. As we established in the last article, 220 Hz, 440 Hz, and 880 Hz are all the note A.) So, with a fundamental frequency of 330 Hz, an E's harmonics should be 660 Hz, 990 Hz, 1320 Hz, 1650 Hz, and 1980 Hz. And if we were to find the fundamental and its first five harmonics on a piano, we'd find they were E, E, B, E, G#, and B. We've got two new notes to play with, and they slot into Julie's "Do-Re-Mi" scheme quite nicely. B fits in as "re," while G# fits at "ti." Now we only have "fa" and "la" left to find.
Where do we find these last two notes? My first inclination would be to work from B, since it's the first harmonic of E that isn't an E. It's a logical thought, but remember that we're building a scale based on the original note of A. Remember how E is the note most closely related to A, because it is the first harmonic of A that isn't an A? Maybe we need to find the note that A is the first non-identical harmonic of. To do this, let's reverse the math. Instead of multiplying 220 Hz by three to get to E, let's divide 220 Hz by three, giving us 73.333 Hz. If we start with 73.333 Hz as our fundamental frequency, its harmonics will be 146.666 Hz, 220 Hz, 293.333 Hz, 366.666 Hz, and 440 Hz. If we were to find those harmonics on a piano, we'd find them at D, D, A, D, F#, and A -- giving us two new notes. Sure enough, D fits at "fa," and F# fits at "la."
This leaves us with our own do, re, mi, fa, so, la, and ti in the notes A, B, C#, D, E, F#, and G#. We now have a major scale, or as Julie puts it, "the tools we use to build a song." These aren't all the notes we have available, though; we also have A#, C, D#, F, and G. We can find them easily enough by calculating out the harmonics of the notes we have; for instance, if we looked at the fundamental and the first five harmonics of a B, we would find they were B, B, F#, B, D#, and F#.
You can hear the major scale being used in everything from "Happy Birthday" to the Ramones' "Blitzkreig Bop." As always, feel free to send me any and all questions and comments you might have 'til next time, when we'll take a look at how different keys are constructed.